Find The Local Maximum And Minimum Values
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Find The Local Maximum And Minimum Values
In mathematics we Find the Local Maximum and Minimum values for the functions.
There are points in the domain of a function where the function does not attain the
greatest value but the values at these points are greater than or less than the values
of the function at the neighboring points.
Such points are called the local minimum and local maximum point. Let us discuss
about the local minimum and local maximum values for the function.
Local maximum:  A function f ( x ) reaches a local maximum at x = a if there is a
region ( a  , a + ) of a such that
f ( x ) < f ( a ) for all x ( a  , a + ), x a
or f ( x )  f ( a ) < 0 for all x ( a  , a + ), x a
In this situation f ( a ) is called the local maximum value of f ( x ) at x = a.
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Local minimum :  A function f ( x ) reaches a local maximum at x = a if there is a
region ( a  , a + ) of a such that
f ( x ) > f ( a ) for all x ( a  , a + ), x a
or f ( x )  f ( a ) > 0 for all x ( a  , a + ), x a
In this situation f ( a ) is called the local minimum value of f ( x ) at x = a.
The points at which a function attains either the local maximum values or local
minimum values are known as the extreme points.
Let's take an example to find all the points of local maximum and local minimum of
the function.
f ( x ) = x3  6x2 + 9x  8
Solution :  Let y = f ( x ) = x3  6x2 + 9x  8. Then,
dy / dx = f' ( x ) = 3x  12x + 9x = 3 ( x2  4x + 3 )
For a local maximum and local minimum, we have
dy / dx = 0 => 3 ( x2  4x + 3 ) = 0 => x = 1, 3.
We have to examine whether these points are points of local maximum or local
minimum or neither of them
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We have dy / dx 3 ( x  1 ) ( x  3 )
The change in signs of dy / dx for different values of x are shown in figure.
Clearly, dy / dx changes sign from positive to negative as increase through 1.
Find the Local Maximum and Minimum valuesp: initial; backgroundcolor: white;
backgroundposition: initial initial; backgroundrepeat: initial initial; "> x = 1 is a point
of a local maximum.
Also, dy / dx changes sign from negative to positive as x increases through 3.
So x = 3 is a point of a local minimum.
We can do some test on the local maximum and local minimum such as first
derivative and higher order derivative test.
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